MathDB

2

Part of 2012 European Mathematical Cup

Problems(2)

GCD's implying GCD's

Source: European Mathematical Cup 2012, Junior Division, Problem 2

7/27/2013
Let SS be the set of positive integers. For any aa and bb in the set we have GCD(a,b)>1GCD(a, b)>1. For any aa, bb and cc in the set we have GCD(a,b,c)=1GCD(a, b, c)=1. Is it possible that SS has 20122012 elements? Proposed by Ognjen Stipetić.
number theorygreatest common divisorfunctionprime numbersnumber theory proposed
Yet another cyclic quadrilateral

Source: European Mathematical Cup 2012, Senior Division, Problem 2

7/27/2013
Let ABCABC be an acute triangle with orthocenter HH. Segments AHAH and CHCH intersect segments BCBC and ABAB in points A1A_1 and C1C_1 respectively. The segments BHBH and A1C1A_1C_1 meet at point DD. Let PP be the midpoint of the segment BHBH. Let D′D' be the reflection of the point DD in ACAC. Prove that quadrilateral APCD′APCD' is cyclic.
Proposed by Matko Ljulj.
geometrygeometric transformationreflectioncircumcirclegeometry proposed